# measure theory – Feynman Path Integral and Wilsonian Renormalization

Everything below is to be viewed in the Euclidean setting with $$d$$ dimensions and all measures are understood to be Borel measures.

The usual problem of Quantum Field Theory is to make sense of Feynman Path Integrals like

$$begin{equation} int exp left(- frac{m^2}{2} int phi^2 + frac{Z}{2} int phi Delta phi – lambda int phi^4 right) mathrm{d} phi end{equation}$$

where the outer integral is supposed to run over some space of test functions.
It is well-known that this does not really work.
What one can do, however, is for $$lambda = 0$$ to produce a “cylinder set probability measure” on e.g the space $$mathcal{S}$$ of Schwartz functions. More generally this works for any continuous positive definite bilinear form on $$mathcal{S}$$ and the resulting cylinder set measure may be pushed forward to $$mathcal{S}’$$ where it miraculously becomes a true Gaussian probability measure. Hence, for such theories we know what we are doing!

With $$lambda neq 0$$ this is no longer the case. In order to ensure the finiteness of expressions of the form $$int phi^4$$ we would like $$phi$$ to live in $$mathcal{S}$$.
To this end, we pick two non-negative mollifiers $$chi, xi in mathcal{D}$$ with $$xi left( 0 right) = 1$$. Furthermore, for any $$epsilon > 0$$ and $$Lambda > 0$$, define

begin{aligned} chi_epsilon left( x right) &= epsilon^{-d} chi left( frac{x}{epsilon} right) \ xi_Lambda left( x right) &= xi left( frac{x}{Lambda} right) \ end{aligned}

and consider the continuous (? – it should certainly be Borel measurable) mapping

begin{aligned} mathcal{M} left( xi_Lambda right) mathcal{C} left( chi_epsilon right) : mathcal{S}’ &to mathcal{S} \ T &mapsto xi_Lambda cdot left( chi_epsilon ast T right) , . end{aligned}

We may now take a Gaussian measure $$mu$$ on $$mathcal{S}’$$ corresponding to some bilinear theory and regularize it as $$left( mathcal{M} left( xi_Lambda right) mathcal{C} left( chi_epsilon right) right)_ast mu$$ to a measure on $$mathcal{S}$$.
One can then define measures

$$begin{equation} nu_{epsilon, Lambda} = exp left( -lambda int phi^4 right) cdot left( mathcal{M} left( xi_Lambda right) mathcal{C} left( chi_epsilon right) right)_ast mu end{equation}$$

normalize them, push them to $$mathcal{S}’$$ again and study their behaviour as $$left( epsilon, Lambda right) to left( 0, infty right)$$.
But this also does not work as it just produces the expected divergences coming from attempting to multiply distributions.
Instead, one has to give up on keeping the model parameters $$m, Z, lambda$$ constant and let them depend on $$epsilon$$ and $$Lambda$$ (i.e on cutoffs) instead. One might then hope that for some nice $$left( epsilon, Lambda right)$$-dependence a limit (or at least a cluster point) $$nu_{0, infty}$$ indeed exists.

From a Wilsonian point of view, we would however like to split this $$nu_{0, infty}$$ into “fast” and “slow” modes depending on $$epsilon$$ and $$Lambda$$. That is, we want to fix a collection $$left( mu_{epsilon, Lambda} right)_{epsilon ge 0, Lambda le infty}$$ of probability measures on $$mathcal{S}’$$ such that for all $$epsilon, delta > 0$$ and all $$Lambda, kappa > 0$$

• $$nu_{0, infty}$$ factorizes over $$mu_{epsilon, Lambda}$$, i.e $$nu_{0, infty} = tilde{nu}_{epsilon, Lambda} ast mu_{epsilon, Lambda}$$ for some probability measure $$tilde{nu}_{epsilon, Lambda}$$ on $$mathcal{S}’$$
• $$mu_{epsilon, Lambda} ast mu_{delta, kappa} = mu_{epsilon+delta, frac{lambda kappa}{Lambda + kappa}}$$ (or someway similar)
• $$lim_{epsilon to 0, Lambda to infty} mu_{epsilon, Lambda} = mu_{0, infty}$$

where somehow $$nu_epsilon$$ and $$tilde{nu}_epsilon$$ should be related.
Then we obtain a renormalization group equation

$$begin{equation} tilde{nu}_{epsilon, Lambda} = tilde{nu}_{epsilon + delta, frac{lambda kappa}{Lambda + kappa}} ast mu_{delta, kappa} end{equation}$$

as well as

$$begin{equation} tilde{nu}_{0, infty} = mu_{0, infty} ast lim_{epsilon to 0, Lambda to infty} tilde{nu}_{epsilon, Lambda} end{equation}$$

But, presuming the existence of some nice QFT corresponding to $$nu_0$$

• Why would such a decomposition with respect to a such a family $$mu_{epsilon, Lambda}$$ exist? i.e why can we separate fast and slow modes in an arbitrary way?
• How do we relate $$nu_{epsilon, Lambda}$$ and $$tilde{nu}_{epsilon, Lambda}$$ i.e how do we understand the map from bare to renormalized quantities and/or vice versa?

PS: The above is pretty much as far as my level of mathematics goes. Please try to not go too much further.